In Figure REF  we compare the PDFs of the longitudinal and transverse components of the velocity gradient predicted by the ML-RDGF model with the DNS results. (Recall that \({a}\)  and \({b}\)  are the phase-space variables conjugate to \({\mathcal {A}}\)  and \({\mathcal {B}}\) , respectively, and we use the subscripts “\(rms\) ” and “\(av\) ” to denote the root-mean-square and averaged values of the variable under consideration). One can see that the PDFs for the longitudinal gradients (Figure REF (a)) are negatively skewed while the PDFs for the transverse gradients are symmetric (Figure REF (b)). The former is associated with the self-amplification of the velocity gradients [1]}, while the latter is a constraint due to isotropy. The predictions from M1 are in excellent agreement with the DNS data, and the model makes realistic predictions at the higher \(_\lambda \)  to which M2 and M3 correspond. The ability of the ML-RDGF model to accurately predict the components of the velocity gradient, and realistic intermittency trends with increasing \(_\lambda \)  (and over a much larger range of \(_\lambda \)  than considered here) were previously demonstrated in detail in the original ML-RDGF paper of [2]}.
